Question

Given the function f(x)=0.5(3)^x, what is the value of f^-1(7)?

Answer 1

`f\left( x \right) =\frac { 1 }{ 2 } \cdot { 3 }^( x )\\ \\ \frac { 1 }{ 2 } \cdot { 3 }^( x )=y`


`\\ \\ 2\cdot \frac { 1 }{ 2 } \cdot { 3 }^( x )=2\cdot y\\ \\ { 3 }^( x )=2y\\ \\ \log _( 3 ){ \left( 2y \right) } =x`


`\\ \\ \therefore \quad { f }^( -1 )\left( x \right) =\log _( 3 ){ \left( 2x \right) }`


`\\ \\ \therefore \quad { f }^( -1 )\left( 7 \right) =\log _( 3 ){ \left( 2\cdot 7 \right) } =\log _( 3 ){ 14 }`